What is the integral from 0 to pi/2 of sin^8(x) ?

The integral from 0 to pi/2 of sin^8(x) is (35pi)/(256)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to pi/2 of sin^8(x)

\int_{0}^{\frac{π}{2}} sin^{8} (x) d x

\frac{35 π}{256}

Decimal Notation

0.42951 \dots

Solution steps

\int_{0}^{\frac{π}{2}} sin^{8} (x) d x

Apply Integral Reduction

= - [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \int_{0}^{\frac{π}{2}} sin^{6} (x) d x

\int_{0}^{\frac{π}{2}} sin^{6} (x) d x = \frac{5 π}{32}

= - [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32}

Simplify

- [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32} : - [\frac{1}{8} sin^{7} (x) cos (x)]_{0}^{\frac{π}{2}} + \frac{35 π}{256}

= - [\frac{1}{8} sin^{7} (x) cos (x)]_{0}^{\frac{π}{2}} + \frac{35 π}{256}

Compute the boundaries:

0

= - 0 + \frac{35 π}{256}

Simplify

= \frac{35 π}{256}

\int_{2}^{5} (3 x + 1) d x

\int_{1}^{\infty} x ln (x) d x

\int_{- 1}^{1} \frac{1}{1 - ∣ x ∣} d x

\int_{1}^{5} x e^{x} d x

\int_{0}^{1} \frac{1}{x ( x ^{2} + 1 )} d x

What is the integral from 0 to pi/2 of sin^8(x) ?
The integral from 0 to pi/2 of sin^8(x) is (35pi)/(256)